Proving Properties of a Group with Every Element of Order 2

[fk378]

Homework Statement

Let G be a group such that every x in G\{e} has order 2.
(a) Let H<=G be a subgroup. Show that for every x in G the subset H U (xH) is also a subgroup.
(b) Show that if G is finite, then |G|=2^n for some integer n.

The Attempt at a Solution

For (a), I know that since x is also part of G, then if multiplied to H (left coset) it will also still be contained in G, but I don't know how to prove it.
For (b), since the order of every element in G is 2, then every set in G has 2 elements, so the order of G is just 2(elements) times how many sets there are...also having trouble proving this.

Please let me know if my train of thoughts are right.

 

[morphism]

Hint for (a): Notice that every element is its own inverse.

Hint for (b): Use part (a).